We study the number of zeros of Abelian integrals for the quadratic centers having almost all their orbits formed by cubics, when we perturb such systems inside the class of all polynomial systems of degree n.
Abstract This paper is devoted to the study of the period function for a class of reversible quadratic system x=-2xy,y=k-1-2kx＋（k＋1）x^2-1/2y^2.We determine the monotonicity of the period function for each k ∈ R....Abstract This paper is devoted to the study of the period function for a class of reversible quadratic system x=-2xy,y=k-1-2kx＋（k＋1）x^2-1/2y^2.We determine the monotonicity of the period function for each k ∈ R. It is proved that the period function has at most one critical point.展开更多
An upper bound B(n)≤ 12{7n+12((-1) n-1)} is derived for the number of zeros of Abelian integralsI(h)=∮ Γ h g(x,y)dy-f(x,y)dx on the open interval (-112,0)∪ (0,+∞), where Γ h is the...An upper bound B(n)≤ 12{7n+12((-1) n-1)} is derived for the number of zeros of Abelian integralsI(h)=∮ Γ h g(x,y)dy-f(x,y)dx on the open interval (-112,0)∪ (0,+∞), where Γ h is the compact component of the algebraic curve H(x,y)=12y 2+13x 3+14x 4=h,f(x,y) and g(x,y) are polynomials of x and y,n= max s{ deg f(x,y), deg g(x,y)} .展开更多
This paper is concerned with limit cycles which bifurcate from a period annulus of a quadratic reversible Lotka-Volterra system with sextic orbits.The authors apply the property of an extended complete Chebyshev syste...This paper is concerned with limit cycles which bifurcate from a period annulus of a quadratic reversible Lotka-Volterra system with sextic orbits.The authors apply the property of an extended complete Chebyshev system and prove that the cyclicity of the period annulus under quadratic perturbations is equal to two.展开更多
The authors investigate a kind of degenerate quadratic Hamiltonian systems with saddle-loop. Under quadratic perturbations, it is proved that the perturbed system has at most two limit cycles in the finite plane. The ...The authors investigate a kind of degenerate quadratic Hamiltonian systems with saddle-loop. Under quadratic perturbations, it is proved that the perturbed system has at most two limit cycles in the finite plane. The proof relies on a careful analysis of a related Abelian integral.展开更多
基金This work was supported by theNational Natural Science Foundation of China (Grant No. 10101031) Guangdong Natural Science Foundations (Grant No. 001289) Natural Science Foundation of Zhongshan University for young teachers.
文摘We study the number of zeros of Abelian integrals for the quadratic centers having almost all their orbits formed by cubics, when we perturb such systems inside the class of all polynomial systems of degree n.
文摘Abstract This paper is devoted to the study of the period function for a class of reversible quadratic system x=-2xy,y=k-1-2kx＋（k＋1）x^2-1/2y^2.We determine the monotonicity of the period function for each k ∈ R. It is proved that the period function has at most one critical point.
文摘An upper bound B(n)≤ 12{7n+12((-1) n-1)} is derived for the number of zeros of Abelian integralsI(h)=∮ Γ h g(x,y)dy-f(x,y)dx on the open interval (-112,0)∪ (0,+∞), where Γ h is the compact component of the algebraic curve H(x,y)=12y 2+13x 3+14x 4=h,f(x,y) and g(x,y) are polynomials of x and y,n= max s{ deg f(x,y), deg g(x,y)} .
基金Project supported by the National Natural Science Foundation of China(Nos.11226152,11201086)the Science and Technology Foundation of Guizhou Province(No.[2012]2167)+1 种基金the Foundation for Distinguished Young Talents in Higher Education of Guangdong(No.2012LYM_0087)the Talent Project Foundation of Guizhou University(No.201104)
文摘This paper is concerned with limit cycles which bifurcate from a period annulus of a quadratic reversible Lotka-Volterra system with sextic orbits.The authors apply the property of an extended complete Chebyshev system and prove that the cyclicity of the period annulus under quadratic perturbations is equal to two.
基金the National Natural Science Foundation of China (No.10101031. No. 10071097). Guangdong Natural Science Foundation (No. 001289)
文摘The authors investigate a kind of degenerate quadratic Hamiltonian systems with saddle-loop. Under quadratic perturbations, it is proved that the perturbed system has at most two limit cycles in the finite plane. The proof relies on a careful analysis of a related Abelian integral.
